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Theorem ccatswrd 12644
Description: Joining two adjacent subwords makes a longer subword. (Contributed by Stefan O'Rear, 20-Aug-2015.)
Assertion
Ref Expression
ccatswrd  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( S substr  <. X ,  Y >. ) concat 
( S substr  <. Y ,  Z >. ) )  =  ( S substr  <. X ,  Z >. ) )

Proof of Theorem ccatswrd
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 swrdcl 12609 . . . . . 6  |-  ( S  e. Word  A  ->  ( S substr  <. X ,  Y >. )  e. Word  A )
21adantr 465 . . . . 5  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( S substr  <. X ,  Y >. )  e. Word  A
)
3 swrdcl 12609 . . . . . 6  |-  ( S  e. Word  A  ->  ( S substr  <. Y ,  Z >. )  e. Word  A )
43adantr 465 . . . . 5  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( S substr  <. Y ,  Z >. )  e. Word  A
)
5 ccatcl 12558 . . . . 5  |-  ( ( ( S substr  <. X ,  Y >. )  e. Word  A  /\  ( S substr  <. Y ,  Z >. )  e. Word  A
)  ->  ( ( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) )  e. Word  A )
62, 4, 5syl2anc 661 . . . 4  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( S substr  <. X ,  Y >. ) concat 
( S substr  <. Y ,  Z >. ) )  e. Word  A )
7 wrdf 12519 . . . 4  |-  ( ( ( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) )  e. Word  A  -> 
( ( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) : ( 0..^ ( # `  (
( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) ) ) --> A )
8 ffn 5731 . . . 4  |-  ( ( ( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) : ( 0..^ ( # `  (
( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) ) ) --> A  ->  ( ( S substr  <. X ,  Y >. ) concat 
( S substr  <. Y ,  Z >. ) )  Fn  ( 0..^ ( # `  ( ( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) ) ) )
96, 7, 83syl 20 . . 3  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( S substr  <. X ,  Y >. ) concat 
( S substr  <. Y ,  Z >. ) )  Fn  ( 0..^ ( # `  ( ( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) ) ) )
10 ccatlen 12559 . . . . . . 7  |-  ( ( ( S substr  <. X ,  Y >. )  e. Word  A  /\  ( S substr  <. Y ,  Z >. )  e. Word  A
)  ->  ( # `  (
( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) )  =  ( ( # `  ( S substr  <. X ,  Y >. ) )  +  (
# `  ( S substr  <. Y ,  Z >. ) ) ) )
112, 4, 10syl2anc 661 . . . . . 6  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( # `  (
( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) )  =  ( ( # `  ( S substr  <. X ,  Y >. ) )  +  (
# `  ( S substr  <. Y ,  Z >. ) ) ) )
12 simpl 457 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  S  e. Word  A
)
13 simpr1 1002 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  X  e.  ( 0 ... Y ) )
14 simpr2 1003 . . . . . . . . . 10  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  Y  e.  ( 0 ... Z ) )
15 simpr3 1004 . . . . . . . . . 10  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  Z  e.  ( 0 ... ( # `  S ) ) )
16 fzass4 11721 . . . . . . . . . . . 12  |-  ( ( Y  e.  ( 0 ... ( # `  S
) )  /\  Z  e.  ( Y ... ( # `
 S ) ) )  <->  ( Y  e.  ( 0 ... Z
)  /\  Z  e.  ( 0 ... ( # `
 S ) ) ) )
1716biimpri 206 . . . . . . . . . . 11  |-  ( ( Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) )  -> 
( Y  e.  ( 0 ... ( # `  S ) )  /\  Z  e.  ( Y ... ( # `  S
) ) ) )
1817simpld 459 . . . . . . . . . 10  |-  ( ( Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) )  ->  Y  e.  ( 0 ... ( # `  S
) ) )
1914, 15, 18syl2anc 661 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  Y  e.  ( 0 ... ( # `  S ) ) )
20 swrdlen 12613 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) ) )  -> 
( # `  ( S substr  <. X ,  Y >. ) )  =  ( Y  -  X ) )
2112, 13, 19, 20syl3anc 1228 . . . . . . . 8  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( # `  ( S substr  <. X ,  Y >. ) )  =  ( Y  -  X ) )
22 swrdlen 12613 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) )  -> 
( # `  ( S substr  <. Y ,  Z >. ) )  =  ( Z  -  Y ) )
2312, 14, 15, 22syl3anc 1228 . . . . . . . 8  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( # `  ( S substr  <. Y ,  Z >. ) )  =  ( Z  -  Y ) )
2421, 23oveq12d 6302 . . . . . . 7  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( # `  ( S substr  <. X ,  Y >. ) )  +  ( # `  ( S substr  <. Y ,  Z >. ) ) )  =  ( ( Y  -  X )  +  ( Z  -  Y ) ) )
25 elfzelz 11688 . . . . . . . . . 10  |-  ( Y  e.  ( 0 ... Z )  ->  Y  e.  ZZ )
2614, 25syl 16 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  Y  e.  ZZ )
2726zcnd 10967 . . . . . . . 8  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  Y  e.  CC )
28 elfzelz 11688 . . . . . . . . . 10  |-  ( X  e.  ( 0 ... Y )  ->  X  e.  ZZ )
2913, 28syl 16 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  X  e.  ZZ )
3029zcnd 10967 . . . . . . . 8  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  X  e.  CC )
31 elfzelz 11688 . . . . . . . . . 10  |-  ( Z  e.  ( 0 ... ( # `  S
) )  ->  Z  e.  ZZ )
3215, 31syl 16 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  Z  e.  ZZ )
3332zcnd 10967 . . . . . . . 8  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  Z  e.  CC )
3427, 30, 33npncan3d 9966 . . . . . . 7  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( Y  -  X )  +  ( Z  -  Y
) )  =  ( Z  -  X ) )
3524, 34eqtrd 2508 . . . . . 6  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( # `  ( S substr  <. X ,  Y >. ) )  +  ( # `  ( S substr  <. Y ,  Z >. ) ) )  =  ( Z  -  X
) )
3611, 35eqtrd 2508 . . . . 5  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( # `  (
( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) )  =  ( Z  -  X ) )
3736oveq2d 6300 . . . 4  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( 0..^ (
# `  ( ( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) ) )  =  ( 0..^ ( Z  -  X
) ) )
3837fneq2d 5672 . . 3  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( ( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) )  Fn  ( 0..^ ( # `  ( ( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) ) )  <->  ( ( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) )  Fn  ( 0..^ ( Z  -  X ) ) ) )
399, 38mpbid 210 . 2  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( S substr  <. X ,  Y >. ) concat 
( S substr  <. Y ,  Z >. ) )  Fn  ( 0..^ ( Z  -  X ) ) )
40 swrdcl 12609 . . . . 5  |-  ( S  e. Word  A  ->  ( S substr  <. X ,  Z >. )  e. Word  A )
4140adantr 465 . . . 4  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( S substr  <. X ,  Z >. )  e. Word  A
)
42 wrdf 12519 . . . 4  |-  ( ( S substr  <. X ,  Z >. )  e. Word  A  -> 
( S substr  <. X ,  Z >. ) : ( 0..^ ( # `  ( S substr  <. X ,  Z >. ) ) ) --> A )
43 ffn 5731 . . . 4  |-  ( ( S substr  <. X ,  Z >. ) : ( 0..^ ( # `  ( S substr  <. X ,  Z >. ) ) ) --> A  ->  ( S substr  <. X ,  Z >. )  Fn  (
0..^ ( # `  ( S substr  <. X ,  Z >. ) ) ) )
4441, 42, 433syl 20 . . 3  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( S substr  <. X ,  Z >. )  Fn  (
0..^ ( # `  ( S substr  <. X ,  Z >. ) ) ) )
45 fzass4 11721 . . . . . . . . 9  |-  ( ( X  e.  ( 0 ... Z )  /\  Y  e.  ( X ... Z ) )  <->  ( X  e.  ( 0 ... Y
)  /\  Y  e.  ( 0 ... Z
) ) )
4645biimpri 206 . . . . . . . 8  |-  ( ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z ) )  ->  ( X  e.  ( 0 ... Z
)  /\  Y  e.  ( X ... Z ) ) )
4746simpld 459 . . . . . . 7  |-  ( ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z ) )  ->  X  e.  ( 0 ... Z ) )
4813, 14, 47syl2anc 661 . . . . . 6  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  X  e.  ( 0 ... Z ) )
49 swrdlen 12613 . . . . . 6  |-  ( ( S  e. Word  A  /\  X  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) )  -> 
( # `  ( S substr  <. X ,  Z >. ) )  =  ( Z  -  X ) )
5012, 48, 15, 49syl3anc 1228 . . . . 5  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( # `  ( S substr  <. X ,  Z >. ) )  =  ( Z  -  X ) )
5150oveq2d 6300 . . . 4  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( 0..^ (
# `  ( S substr  <. X ,  Z >. ) ) )  =  ( 0..^ ( Z  -  X ) ) )
5251fneq2d 5672 . . 3  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( S substr  <. X ,  Z >. )  Fn  ( 0..^ (
# `  ( S substr  <. X ,  Z >. ) ) )  <->  ( S substr  <. X ,  Z >. )  Fn  ( 0..^ ( Z  -  X ) ) ) )
5344, 52mpbid 210 . 2  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( S substr  <. X ,  Z >. )  Fn  (
0..^ ( Z  -  X ) ) )
54 simpr 461 . . . . 5  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Z  -  X ) ) )  ->  x  e.  ( 0..^ ( Z  -  X ) ) )
5526, 29zsubcld 10971 . . . . . 6  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( Y  -  X )  e.  ZZ )
5655adantr 465 . . . . 5  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Z  -  X ) ) )  ->  ( Y  -  X )  e.  ZZ )
57 fzospliti 11825 . . . . 5  |-  ( ( x  e.  ( 0..^ ( Z  -  X
) )  /\  ( Y  -  X )  e.  ZZ )  ->  (
x  e.  ( 0..^ ( Y  -  X
) )  \/  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) ) )
5854, 56, 57syl2anc 661 . . . 4  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Z  -  X ) ) )  ->  ( x  e.  ( 0..^ ( Y  -  X ) )  \/  x  e.  ( ( Y  -  X
)..^ ( Z  -  X ) ) ) )
592adantr 465 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Y  -  X ) ) )  ->  ( S substr  <. X ,  Y >. )  e. Word  A )
604adantr 465 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Y  -  X ) ) )  ->  ( S substr  <. Y ,  Z >. )  e. Word  A )
6121oveq2d 6300 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( 0..^ (
# `  ( S substr  <. X ,  Y >. ) ) )  =  ( 0..^ ( Y  -  X ) ) )
6261eleq2d 2537 . . . . . . . 8  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( x  e.  ( 0..^ ( # `  ( S substr  <. X ,  Y >. ) ) )  <-> 
x  e.  ( 0..^ ( Y  -  X
) ) ) )
6362biimpar 485 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Y  -  X ) ) )  ->  x  e.  ( 0..^ ( # `  ( S substr  <. X ,  Y >. ) ) ) )
64 ccatval1 12560 . . . . . . 7  |-  ( ( ( S substr  <. X ,  Y >. )  e. Word  A  /\  ( S substr  <. Y ,  Z >. )  e. Word  A  /\  x  e.  (
0..^ ( # `  ( S substr  <. X ,  Y >. ) ) ) )  ->  ( ( ( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) `  x )  =  ( ( S substr  <. X ,  Y >. ) `  x
) )
6559, 60, 63, 64syl3anc 1228 . . . . . 6  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Y  -  X ) ) )  ->  ( (
( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) `  x )  =  ( ( S substr  <. X ,  Y >. ) `
 x ) )
66 simpll 753 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Y  -  X ) ) )  ->  S  e. Word  A )
67 simplr1 1038 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Y  -  X ) ) )  ->  X  e.  ( 0 ... Y
) )
6819adantr 465 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Y  -  X ) ) )  ->  Y  e.  ( 0 ... ( # `
 S ) ) )
69 simpr 461 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Y  -  X ) ) )  ->  x  e.  ( 0..^ ( Y  -  X ) ) )
70 swrdfv 12614 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) ) )  /\  x  e.  ( 0..^ ( Y  -  X
) ) )  -> 
( ( S substr  <. X ,  Y >. ) `  x
)  =  ( S `
 ( x  +  X ) ) )
7166, 67, 68, 69, 70syl31anc 1231 . . . . . 6  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Y  -  X ) ) )  ->  ( ( S substr  <. X ,  Y >. ) `  x )  =  ( S `  ( x  +  X
) ) )
7265, 71eqtrd 2508 . . . . 5  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Y  -  X ) ) )  ->  ( (
( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) `  x )  =  ( S `  ( x  +  X
) ) )
732adantr 465 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( S substr  <. X ,  Y >. )  e. Word  A
)
744adantr 465 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( S substr  <. Y ,  Z >. )  e. Word  A
)
7521, 35oveq12d 6302 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( # `  ( S substr  <. X ,  Y >. ) )..^ ( ( # `  ( S substr  <. X ,  Y >. ) )  +  (
# `  ( S substr  <. Y ,  Z >. ) ) ) )  =  ( ( Y  -  X )..^ ( Z  -  X ) ) )
7675eleq2d 2537 . . . . . . . 8  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( x  e.  ( ( # `  ( S substr  <. X ,  Y >. ) )..^ ( (
# `  ( S substr  <. X ,  Y >. ) )  +  ( # `  ( S substr  <. Y ,  Z >. ) ) ) )  <->  x  e.  (
( Y  -  X
)..^ ( Z  -  X ) ) ) )
7776biimpar 485 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  x  e.  ( ( # `  ( S substr  <. X ,  Y >. ) )..^ ( (
# `  ( S substr  <. X ,  Y >. ) )  +  ( # `  ( S substr  <. Y ,  Z >. ) ) ) ) )
78 ccatval2 12561 . . . . . . 7  |-  ( ( ( S substr  <. X ,  Y >. )  e. Word  A  /\  ( S substr  <. Y ,  Z >. )  e. Word  A  /\  x  e.  (
( # `  ( S substr  <. X ,  Y >. ) )..^ ( ( # `  ( S substr  <. X ,  Y >. ) )  +  ( # `  ( S substr  <. Y ,  Z >. ) ) ) ) )  ->  ( (
( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) `  x )  =  ( ( S substr  <. Y ,  Z >. ) `
 ( x  -  ( # `  ( S substr  <. X ,  Y >. ) ) ) ) )
7973, 74, 77, 78syl3anc 1228 . . . . . 6  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( ( ( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) `  x )  =  ( ( S substr  <. Y ,  Z >. ) `  (
x  -  ( # `  ( S substr  <. X ,  Y >. ) ) ) ) )
80 simpll 753 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  S  e. Word  A
)
81 simplr2 1039 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  Y  e.  ( 0 ... Z ) )
82 simplr3 1040 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  Z  e.  ( 0 ... ( # `  S ) ) )
8321oveq2d 6300 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( x  -  ( # `  ( S substr  <. X ,  Y >. ) ) )  =  ( x  -  ( Y  -  X ) ) )
8483adantr 465 . . . . . . . 8  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( x  -  ( # `  ( S substr  <. X ,  Y >. ) ) )  =  ( x  -  ( Y  -  X ) ) )
8534oveq2d 6300 . . . . . . . . . . 11  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( Y  -  X )..^ ( ( Y  -  X
)  +  ( Z  -  Y ) ) )  =  ( ( Y  -  X )..^ ( Z  -  X
) ) )
8685eleq2d 2537 . . . . . . . . . 10  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( x  e.  ( ( Y  -  X )..^ ( ( Y  -  X )  +  ( Z  -  Y
) ) )  <->  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) ) )
8786biimpar 485 . . . . . . . . 9  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  x  e.  ( ( Y  -  X
)..^ ( ( Y  -  X )  +  ( Z  -  Y
) ) ) )
8832, 26zsubcld 10971 . . . . . . . . . 10  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( Z  -  Y )  e.  ZZ )
8988adantr 465 . . . . . . . . 9  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( Z  -  Y )  e.  ZZ )
90 fzosubel3 11845 . . . . . . . . 9  |-  ( ( x  e.  ( ( Y  -  X )..^ ( ( Y  -  X )  +  ( Z  -  Y ) ) )  /\  ( Z  -  Y )  e.  ZZ )  ->  (
x  -  ( Y  -  X ) )  e.  ( 0..^ ( Z  -  Y ) ) )
9187, 89, 90syl2anc 661 . . . . . . . 8  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( x  -  ( Y  -  X
) )  e.  ( 0..^ ( Z  -  Y ) ) )
9284, 91eqeltrd 2555 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( x  -  ( # `  ( S substr  <. X ,  Y >. ) ) )  e.  ( 0..^ ( Z  -  Y ) ) )
93 swrdfv 12614 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  Y  e.  (
0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) )  /\  ( x  -  ( # `
 ( S substr  <. X ,  Y >. ) ) )  e.  ( 0..^ ( Z  -  Y ) ) )  ->  (
( S substr  <. Y ,  Z >. ) `  (
x  -  ( # `  ( S substr  <. X ,  Y >. ) ) ) )  =  ( S `
 ( ( x  -  ( # `  ( S substr  <. X ,  Y >. ) ) )  +  Y ) ) )
9480, 81, 82, 92, 93syl31anc 1231 . . . . . 6  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( ( S substr  <. Y ,  Z >. ) `
 ( x  -  ( # `  ( S substr  <. X ,  Y >. ) ) ) )  =  ( S `  (
( x  -  ( # `
 ( S substr  <. X ,  Y >. ) ) )  +  Y ) ) )
9583oveq1d 6299 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( x  -  ( # `  ( S substr  <. X ,  Y >. ) ) )  +  Y )  =  ( ( x  -  ( Y  -  X )
)  +  Y ) )
9695adantr 465 . . . . . . . 8  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( ( x  -  ( # `  ( S substr  <. X ,  Y >. ) ) )  +  Y )  =  ( ( x  -  ( Y  -  X )
)  +  Y ) )
97 elfzoelz 11797 . . . . . . . . . . 11  |-  ( x  e.  ( ( Y  -  X )..^ ( Z  -  X ) )  ->  x  e.  ZZ )
9897zcnd 10967 . . . . . . . . . 10  |-  ( x  e.  ( ( Y  -  X )..^ ( Z  -  X ) )  ->  x  e.  CC )
9998adantl 466 . . . . . . . . 9  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  x  e.  CC )
10027, 30subcld 9930 . . . . . . . . . 10  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( Y  -  X )  e.  CC )
101100adantr 465 . . . . . . . . 9  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( Y  -  X )  e.  CC )
10227adantr 465 . . . . . . . . 9  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  Y  e.  CC )
10399, 101, 102subadd23d 9952 . . . . . . . 8  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( ( x  -  ( Y  -  X ) )  +  Y )  =  ( x  +  ( Y  -  ( Y  -  X ) ) ) )
10427, 30nncand 9935 . . . . . . . . . 10  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( Y  -  ( Y  -  X
) )  =  X )
105104oveq2d 6300 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( x  +  ( Y  -  ( Y  -  X )
) )  =  ( x  +  X ) )
106105adantr 465 . . . . . . . 8  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( x  +  ( Y  -  ( Y  -  X )
) )  =  ( x  +  X ) )
10796, 103, 1063eqtrd 2512 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( ( x  -  ( # `  ( S substr  <. X ,  Y >. ) ) )  +  Y )  =  ( x  +  X ) )
108107fveq2d 5870 . . . . . 6  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( S `  ( ( x  -  ( # `  ( S substr  <. X ,  Y >. ) ) )  +  Y
) )  =  ( S `  ( x  +  X ) ) )
10979, 94, 1083eqtrd 2512 . . . . 5  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( ( ( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) `  x )  =  ( S `  ( x  +  X ) ) )
11072, 109jaodan 783 . . . 4  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  ( x  e.  ( 0..^ ( Y  -  X ) )  \/  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) ) )  ->  ( (
( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) `  x )  =  ( S `  ( x  +  X
) ) )
11158, 110syldan 470 . . 3  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Z  -  X ) ) )  ->  ( (
( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) `  x )  =  ( S `  ( x  +  X
) ) )
112 simpll 753 . . . 4  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Z  -  X ) ) )  ->  S  e. Word  A )
11348adantr 465 . . . 4  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Z  -  X ) ) )  ->  X  e.  ( 0 ... Z
) )
114 simplr3 1040 . . . 4  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Z  -  X ) ) )  ->  Z  e.  ( 0 ... ( # `
 S ) ) )
115 swrdfv 12614 . . . 4  |-  ( ( ( S  e. Word  A  /\  X  e.  (
0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) )  /\  x  e.  ( 0..^ ( Z  -  X
) ) )  -> 
( ( S substr  <. X ,  Z >. ) `  x
)  =  ( S `
 ( x  +  X ) ) )
116112, 113, 114, 54, 115syl31anc 1231 . . 3  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Z  -  X ) ) )  ->  ( ( S substr  <. X ,  Z >. ) `  x )  =  ( S `  ( x  +  X
) ) )
117111, 116eqtr4d 2511 . 2  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Z  -  X ) ) )  ->  ( (
( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) `  x )  =  ( ( S substr  <. X ,  Z >. ) `
 x ) )
11839, 53, 117eqfnfvd 5978 1  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( S substr  <. X ,  Y >. ) concat 
( S substr  <. Y ,  Z >. ) )  =  ( S substr  <. X ,  Z >. ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   <.cop 4033    Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6284   CCcc 9490   0cc0 9492    + caddc 9495    - cmin 9805   ZZcz 10864   ...cfz 11672  ..^cfzo 11792   #chash 12373  Word cword 12500   concat cconcat 12502   substr csubstr 12504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-fzo 11793  df-hash 12374  df-word 12508  df-concat 12510  df-substr 12512
This theorem is referenced by:  wrdcctswrd  12653  swrdccatwrd  12656  wrdeqcats1  12662  wrdeqs1cat  12663  splid  12692  splval2  12696  swrds2  12846  efgredleme  16567  efgredlemc  16569  efgcpbllemb  16579  frgpuplem  16596  wrdsplex  28163
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